Mix together the ground beef, egg, dry bread crumbs, salt, and pepper.
Divide meat mixture into 6 equal portions, forming each portion into a patty about V2 inch thick and 4 inches in diameter.
Mold each patty evenly over the outside of a muffin cup.
Place on a barbecue grill, 4 to 6 inches above glowing coals, with the inside of the muffin cup facing down.
Cook until bottom edges are browned and the top part of the mound is still slightly pink.
(Meat next to cup should be cooked enough that it will hold the shape of a cup when turned.) Turn; remove foil cups carefully.
Fill meat cups with the Sauteed Mushrooms; top each evenly with the shredded cheese; cook only until cheese melts